Let `aa n db` be the coefficients of `x^3` in `(1+x+2x^2+3x^2)^4a n d(1+x+2x^2+3x^3+4x^4)^4,` then respectively. Then the value of `4a//b` is.

Explanation

To find the value of \( \frac{4a}{b} \), where \( a \) and \( b \) are the coefficients of \( x^3 \) in the expressions \( (1+x+2x^2+3x^3)^4 \) and \( (1+x+2x^2+3x^3+4x^4)^4 \) respectively, we can follow these steps:

### Step 1: Find the coefficient \( a \) in \( (1+x+2x^2+3x^3)^4 \)

To find \( a \), we need to determine the coefficient of \( x^3 \) in the expansion of \( (1+x+2x^2+3x^3)^4 \).

Using the multinomial expansion, we can express the coefficient of \( x^3 \) as follows:

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